1. Field of the Invention
The present invention relates to groundwater remediation. In particular, this invention relates to a method and apparatus for remediating a contaminated groundwater site using piecewise continuous control.
2. Background Art
Groundwater contamination has become a significant problem posing a threat to cleanliness of natural groundwater supplies. Heretofore, many methods and apparatus have been proposed to identify remediation (decontamination or stabilization) plans of groundwater supplies. These methods and apparatus can be quite complex. Given the complexity of the methods and apparatus to remediate groundwater supplies, computer model approaches have increased in importance in proposing best or "optimal" remediation plans. Each of these approaches for proposing optimal remediation plans, however, has certain disadvantages.
A. Heuristic Approaches: PA0 B. Linear Approaches and Nonlinear Gradient Approaches: PA0 C. Dynamic Control Approaches:
Some computer model approaches use approximations or heuristics in proposing various remediation plans. These approaches include simulated annealing.sup.1,2, genetic algorithms.sup.3,4, and artificial neural networks.sup.5. Heuristic rules are used to eliminate plan alternatives, which in turn simplify the search for a remediation plan.
Heuristic approaches can address problems which have discontinuous controls and discontinuous objectives. A discontinuous control must have a value selected from a plurality of predetermined discrete values. In contrast, a continuous control may have any value selected from a continuous interval of values between an upper and lower limit. A discontinuous objective often results when discontinuous controls are considered. For example, if only five discrete sizes (discontinuous control) for a given remediation device are allowed and the objective is to minimize cost, then the result is five discrete "fixed costs" (discontinuous objective) wherein each fixed cost corresponds to each remediation device. The ability to solve problems with discontinuous objectives is important when considering an objective such as fixed costs.
A disadvantage of the heuristic approaches, however, is that they cannot guarantee that an optimal or best solution will be found, but only that a "good" solution is likely to be found. One of the eliminated designs could be the optimal design. The simulated annealing approach, for example, for groundwater remediation.sup.1,2 is a heuristic, probablistic method for large scale combinatorial optimization problems. Simulated annealing involves a random search designed to find a good solution, and therefore, cannot guarantee an optimal solution. Moreover, it has been applied to groundwater management.sup.1 in circumstances where there are only a few design alternatives to choose from. The genetic algorithm approach.sup.3,4 for groundwater remediation is similar to the simulated annealing approach in that it is also a heuristic, probablistic method, and uses a random search. Again, an optimal solution is not guaranteed. The artificial neural network approach.sup.5 constructs a neural network to approximate a groundwater model's response to different remediation designs. A genetic algorithm is then used with the neural network approximation to select a design. The advantage of this approach over the aforementioned genetic algorithm approach is that the computational effort of the groundwater model simulation is reduced. A serious disadvantage, however, is that the multiple levels of an approximation it requires render this approach highly imprecise. Moreover, all three heuristic approaches are computationally inefficient when extended to dynamic control problems.
Dynamic control problems allow the design to change over time. For example, the pumping rate of an extraction well may have to be changed over time to account for changes to the condition of the site. Under such circumstances, the three heuristic approaches typically require a geometric increase in computational time to solve such problems. Geometrically increasing effort severely limits the problem size which can be solved.
Several linear approaches have been proposed in the related art .sup.7,20,21,22,23 for remediation design. Because the groundwater flow-contamination transport system response is non-linear, however, linear optimization approaches are not precise in representing the actual design problems involved in groundwater remediation processes. Accordingly, additional approximations resulting in imprecision are often required to account for the nonlinearity.sup.7.
Nonlinear gradient approaches, on the other hand, are more precise than heuristic approaches. Mathematical theory guarantee optimal solutions are found for certain classes of problems. The only approximation required in nonlinear gradient approaches is approximation of the characteristics of an actual groundwater system when creating a groundwater model. A disadvantage of nonlinear gradient approaches, however, is that the objectives considered must be a continuous function of the design variables. Fixed cost is a discrete objective and thus is difficult, although not impossible, to accomplish.sup.11,29 (see Appendix C.)
A nonlinear outer approximation type optimization procedure has been used and adapted to "pump and treat" remediation designs to approximate fixed costs.sup.29. This approach, however, has limited application in that it considers only a few potential well sites, and the computational work of this approach grows geometrically with each additional well location that is considered. This is a clear disadvantage because it is desirable to consider as many locations as possible to ensure a highly efficient method is found.
A nonlinear gradient approach proposed.sup.14,26,27,28 in the related art is the nonlinear program package MINOS.sup.12 which is used with the finite element model SUTRA.sup.13. The combination of these two preexisting computer programs results in an optimal groundwater design program. The approach finds a single, fixed rate (nondynamic) pumping strategy solution for "pump and treat" groundwater remediation systems. A disadvantage of this approach, however, is that it only provides a nondynamic solution. While the MINOS-SUTRA approach can be extended to solve problems which allow the pumping rates to change once during the cleanup.sup.8, it is more problematic in solving problems which allow more than one pumping change (more control dynamics) due to MINOS requiring geometrically more effort with each additional dynamic degree of freedom involved in the analysis.
Another type of nonlinear gradient approach is optimal control analysis. Optimal control analysis was specifically developed to solve dynamic problems efficiently. Optimal control analysis takes advantage of the law of cause and effect (a decision made in the future cannot affect the present) to solve optimal control problems efficiently.
The groundwater remediation approaches, as discussed above, do not allow the cleanup strategy to change over time. However, time-varying strategies can remove more of the contamination for less cost.sup.8,9,10. This is because during a cleanup, the spatial extent of the contaminated volume will change. What may have been the most efficient operation and location of the remediation devices in the beginning, may not be the most efficient later. Dynamic systems adapted to the movement and reduction of the contamination over time to maintain high efficiency throughout the remediation process. The savings in remediation operation costs using a dynamic system can be as large as 50%. Thus, if a remediation project is projected to cost several millions of dollars, then savings will also likely be on the order of millions of dollars.
To achieve these savings, the identification of optimal dynamic designs must be made computationally practical for large, three-dimensional problems. Optimal dynamic designs are more difficult to compute than nondynamic designs because there are more degrees of freedom involved. Several optimal dynamic approaches to groundwater remediation have been proposed in the related art. The MINOS-SUTRA method.sup.8 already mentioned is a quasi-dynamic approach. Nevertheless, unlike other optimal dynamic approaches, the MINOS-SUTRA method does not take advantage of the temporal law of cause and effect for computational efficiency.
All other dynamic optimal control approaches for groundwater remediation are based on either the optimal method called discrete-time.sup.15 differential dynamic programming (dDDP), or continuous-time.sup.16 differential dynamic programming (cDDP). The first discrete-time methods, dDDP, to groundwater management.sup.24 did not consider the cleanup of groundwater contamination. This approach was originally intended to be used only for managing water supply from an unconfined aquifer. In addition, the first dDDP method involving contamination used a one-dimensional finite difference groundwater model.sup.17. This approach, however, incorporated certain types of model parameter uncertainty. The approach was later extended to a small two-dimensional finite difference model for groundwater.sup.18. Applications of dDDP using a two-dimensional finite element groundwater model.sup.19 and a two-dimensional bioremediation model.sup.30 have been developed, but were simplified by using successive linear approximations with the groundwater model. These approximations are referred to as Sequential Approximation Linear Quadratic Regulator (SALQR). Even with these simplifications, however, these approaches are too computationally demanding for large, realistic groundwater models.
The dDDP-calculated optimal policies can also be modified with a nonlinear weighted feedback law as disclosed in U.S. Pat. No. 5,468,088 (which is hereby incorporated by reference) to maintain remediation efficiency even when there is an error in the groundwater model predictions. This feedback approach is limited, however, to problem sizes which can be solved using discrete-time dDDP approaches.sup.9,10,19. Another approach, called the (discrete-time) management period method.sup.10, reduces the computational demand of applying dDDP (SALQR), but it achieves this by reducing the dynamic degrees of freedom that are available. This puts two important objectives at odds. Finding the most efficient remediation method for a particular site is often the most important objective. This objective is best accomplished by increasing the dynamic degrees of freedom that are available. This objective is at odds with the objective of reducing the computational effort to a practical level by decreasing the dynamic degrees of freedom available.
Another approach proposed to reduce the computational demand of dDDP is the Quasi-Newton approximation method.sup.19, which reduces the computational effort by about 50%. This reduction is, however, not sufficient to enable the method to consider large realistic groundwater models.
Another dynamic optimal control approach, cDDP, as discussed above, is based on continuous-time optimal control, rather than discrete-time. Continuous-time methods treat time as a continuous variable. This is in contrast to discrete-time methods such as dDDP which treat time as a discrete variable. When dDDP is applied, a time step must be selected over which the remediation policy must remain fixed. The cDDP approach, on the other hand, assumes the remediation system can change continually over time, which leads to a maximally dynamic system design.
One advantage of cDDP over dDDP/SALQR is that the interface between cDDP and a given groundwater model is simpler than with dDDP/SALQR. This is because the continuous-time aspect of cDDP results in simpler derivative calculations. Simpler derivatives make changing the underlying groundwater model simpler. This is important because underlying groundwater models may need to be changed and updated frequently whenever technology or regulatory requirements change. Another advantage of cDDP is that even though the control may change continuously (maximally dynamic), the computational effort is not infinite. This is in contrast to dDDP/SALQR, which requires an infinite amount of computation for maximally dynamic problems. The cDDP method tracks control trends or rates of change rather than single control values. This difference manifests itself in cDDP's integration of ordinary differential equations, as opposed to dDDP/SALQR's step by step linear algebraic procedure. The main disadvantages of the cDDP method, however, is that it requires large amounts of computer memory and it is not always mechanically feasible or desirable to continuously adjust the remediation devices (such as pumping wells or nutrient injection systems) over time.
What is needed, therefore, is a dynamic groundwater remediation approach that achieves the best properties of both dDDP/SALQR and cDDP, while being more flexible and less computationally demanding than either one.
SUMMARY OF THE INVENTION
The present invention is a method and apparatus for decontaminating or stablizing a contaminated groundwater site using piecewise continuous control (i.e., monitoring and making appropriate changes in the cleanup of the site either continuously in time or at discrete times). The method steps of this invention include, determining contaminated groundwater site characterization information; generating a model of the contaminated groundwater site from the characterization information; designing a remediation system for the model using piecewise continuous control; and implementing the remediation system to decontaminate or stabilize the contaminated groundwater site.
The site characterization information includes types of contaminants (e.g., petroleum waste products, polychlorinated biphenyls (PCBs)) and location of the contaminants at the contaminated groundwater site. In addition, the site characterization information includes physical properties of the contaminated groundwater site including depth, topography, volume, materials, hydraulic conductivity, porosity, dispersivity, retardation, biological reactivity, hydraulic head, and recharge rates.
The present invention includes selecting and generating a model such as a finite element model, finite difference model, finite volume model, spectral method model, boundary element model, transform model, neural network model, or analytic model. Next, state variables such as hydraulic heads, nutrient concentrations, biomass and contamination concentrations; and system design variables such as location of remediation devices, operational rate of remediation devices, and inflow concentrations of nutrients or surfactants into remediation devices are implemented into the model. After state variables and system design variables are selected, design objectives such as minimizing cost, maximizing efficiency, minimizing risk, minimizing cleanup time, minimizing contamination level, minimizing number of cleanup devices, and minimizing flow rate from cleanup devices are selected. Next, design constraints (physical, biological, regulatory, and/or economic) such as pumping rate ranges for all extraction wells; pumping rate ranges for all injection wells; a hydraulic capture zone for stability of the contamination (stops it from spreading); a cleanup level (i.e. the cleanliness of the groundwater at the end of remediation); and a maximum water treatment plant size are specified. Once the objective function and constraints are selected, the constraints are incorporated into the objective function(s), parameters are initialized, and an initial arbitrary remediation system design is provided. Piecewise continuous control equations are integrated and a design update law is established. The updated designs are then simulated and tested for a satisfactory improvement in the objective function (e.g, the objective function approaching zero). The piecewise continuous control process using the improved designs is repeated if an optimal design is not determined.
The present invention is designed using devices such as extraction wells, injection wells, observation wells, slurry walls, infiltration caps, soil vapor extractors, air spargers, bioventors, and water treatment plants. One type of device may stimulate biodegradation of contamination in the contaminated groundwater site.
The present invention also includes an apparatus having a processing unit (e.g., central processing unit (CPU)), a memory system, and a program stored in the memory system for execution on the processing unit, the program including: a model of a contaminated groundwater site; and a piecewise continuous control optimization mechanism that optimizes remediation of the contaminated groundwater site over time.
The present invention in a different aspect includes a program product having a recordable media (e.g., Read Only Memory, Compact Disc), and a program recorded on the recordable media accessible by a computer system for execution on a central processing unit. The program includes a piecewise continuous control mechanism that optimizes remediation of a contaminated groundwater site over time.
The present invention has been tested using both a two-dimensional finite difference and a three-dimensional finite element model of groundwater flow and contamination transport. These tests demonstrate that the invention can identify optimal remediation designs given a user specified objective function and additional constraints. In addition, any twice continuously differentiable objective function can be used with the present invention. For example, if the objective function is selected to represent system cost, then the method will result in a low cost system. Moreover, the number or dynamic degrees of freedom is selected without regard to the computational effort required. This is possible because, in the present invention, unlike previous approaches, the computational effort is only weakly dependent on the user selected dynamic degrees of freedom available to the control. Unlike the related art management period method, which is a discrete-time method, the present invention is a continuous-time management period method.
The present invention provides a dynamic remediation approach that is highly cost efficient. This advantage satisfies a recognized existing need to reduce remediation costs. This will not only enable more contaminated sites to be detoxified, but the success rate for each is likely to be improved. Dynamic remediation systems have been shown to be more cost efficient than nondynamic systems.sup.8,9,10. The present invention also allows greater freedom in selecting the form of the dynamic policy than other time-varying methods in the related art.
The present invention provides a dynamic remediation design procedure which has the advantages of both the related approaches dDDP/SALQR and cDDP, without their respective disadvantages. Like cDDP and unlike dDDP/SALQR, the present invention has the advantage of treating time as a continuous process. Unlike cDDP, however, and like dDDP/SALQR, the present invention provides management periods. The present invention's management periods are also more flexible than dDDP/SALQR's management periods. Unlike dDDP/SALQR, the present invention allows continuous and discontinuous dynamic changes to the design during each management period. This advantage of the present invention gives the user greater flexibility to maximize efficiency and obey mechanical constraints. The present invention also requires less computational effort and memory usage than either cDDP or dDDP/SALQR. The present invention provides a continuous-time optimal design procedure which is simpler.sup.25 to interface with groundwater models than discrete-time procedures. This aspect is useful because it is anticipated that groundwater models will continually improve the change over time. Also, regulatory requirements which specify what groundwater model(s) may be changed from time to time and from region to region. The ease of interfacing the present invention to new models is an advantage when new or different models are required. Further, it is easier.sup.25 to exploit groundwater model sparsity with continuous-time control.
The present invention provides mechanically feasible designs and has the numerical advantages of a continuous-time optimal control method. The present invention provides an advantage over the related art continuous-time procedure cDDP because the design is characterized by piecewise continuous functions of time. For example, the design engineer has the freedom to require the optimal remediation device operation rates to be constant, or varied over simple functions period to satisfy mechanical constraints.
The present invention provides precise optimal solutions characteristic of nonlinear gradient methods. This is an advantage over all heuristic and probabilistic methods in the related art because these methods do not guarantee optimal designs. Also, the present invention has the advantage over linear optimization methods because it is a fully nonlinear method. Non-linear methods directly incorporate the nonlinearity inherent in groundwater contamination system response.
The present invention provides a design procedure for dynamic remediation systems which is much less computationally demanding than related art approaches. Also, the present invention provides a remediation design approach which incorporates fully three dimensional groundwater flow and transport models. Most related art approaches are two dimensional.
The present invention allows more freedom in specifying the objective (or objective function) than related art optimal control approaches. This added freedom insures that the most appropriate objective can be specified for each specific remediation project.
The present invention provides a design procedure with a high degree of flexibility of allowing design variable dynamics. The flexibility of design dynamics (management periods) allows both mechanical constraints and optimum cleanup efficiency to be realized. The present invention is also less computationally sensitive to the design dynamics than the related art. The computational effort associated with increasing the dynamic freedom of the design is less than that associated with any related art method.
The present invention provides dynamic or time-changing remediation systems which have been shown to be more cost efficient than nondynamic systems.sup.8,9,10.
The present invention does not require that the dynamic freedom be selected to make the computation feasible on a computer as can be the case with all other methods in the related art. Another advantage of the present invention is that it provides a continuous-time (as opposed to discrete or static) optimization process which is easily interfaced with improved groundwater models when they become available. It provides a continuous-time optimization process which can take advantage of adaptive precision ODE integrators and sparsity to maximize computational efficiency.
Another advantage of the present invention is that it provides a precise nonlinear gradient method which does not yield "trickle" or low flow rate devices or wells in the optimal design because the poor cost efficiency of trickle devices can be correctly represented (Appendix C).
Further advantages of the present invention will become apparent from a consideration of the following drawings and detailed description.